Remarks On The Regular Group Representation
The identity group element corresponds to the identity permutation. All other group elements correspond to a permutation that does not leave any element unchanged. Since this also applies for powers of a group element, lower than the order of that element, each element corresponds to a permutation which consists of cycles which are of the same length: this length is the order of that element. The elements in each cycle form a left coset of the subgroup generated by the element.
Read more about this topic: Cayley's Theorem
Famous quotes containing the words remarks, regular and/or group:
“An illustrious individual remarks that Mrs. [Elizabeth Cady] Stanton is the salt, Anna Dickinson the pepper, and Miss [Susan B.] Anthony the vinegar of the Female Suffrage movement. The very elements get the “white male” into a nice pickle.”
—Anonymous, U.S. women’s magazine contributor. The Revolution (August 19, 1869)
““I couldn’t afford to learn it,” said the Mock Turtle with a sigh. “I only took the regular course.”
“What was that?” inquired Alice.
“Reeling and Writhing, of course, to begin with,” the Mock Turtle replied; “and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
“I never heard of ‘Uglification,’” Alice ventured to say.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“...Women’s Studies can amount simply to compensatory history; too often they fail to challenge the intellectual and political structures that must be challenged if women as a group are ever to come into collective, nonexclusionary freedom.”
—Adrienne Rich (b. 1929)